[FOM] Re: The rule of generalization in FOL

Timothy Y. Chow tchow at alum.mit.edu
Wed Sep 8 10:21:01 EDT 2004

```Arnon Avron wrote:
>there is no single "First Order Logic", but there are TWO different ones.

This reminds me of something that's been bothering me lately.  I've been
reading Stephen Cook's excellent course notes for CSC 2429H: "Proof
complexity and bounded arithmetic."  He gives a version of Gentzen's
systems PK and LK.  What surprises me is that one (or two, depending
on whether you think forall is an abbreviation for ~exists~ or not) of
the LK sequent calculus rules does *not* preserve logical consequence,
as defined a few pages earlier.

The rule in question is this.  Let X and Y denote arbitrary finite
sequences of first-order formulas.  Let A(b) denote a formula with free
variable b.  Then we have the rule

X  ->  Y, A(b)
------------------------
X  ->  Y, forall x: A(x)

provided that b does not occur free in Y.  On the other hand, "phi is a
logical consequence of psi" is defined to hold iff for all structures M
and all interpretations sigma, if M |= psi[sigma] then M |= phi[sigma].

Working with this system makes my head spin because I have to keep reining
in my natural urge to interchange derivability and logical consequence.
My question is, what is the compensating advantage of this system that
makes this apparent mismatch worthwhile?

Tim

```